Concave Lenses: Focal
Point

In this section we deal with a very important aspect of
concave lenses, namely focal point. Many people mistakenly
think that the place where images are formed is the focal point.
Rather a very specific definition is used to define this important
property of concave lenses.

First, a concave lens is one which is thinner in the center
than it is near the edges. This is shown in this diagram:

When we look at the cross-section of a concave lens we notice
that the edges resemble prisms. In fact, a stack of prisms of
varying angles can be used to simulate the actions of a concave
lens. One such is shown here and is called a Fresnel
Lens.

Light passing through the angled prisms near the edges is bent
significantly while light passing through the flat, central area
is hardly bent at all. Light rays which are parallel to one
another when approaching such an arrangement are spread out
becoming diverging as shown here:

We start with this general pattern to define the focal point
for our concave lens.

DEFINITION 1:

The focal point of a concave lens is the point where light rays
parallel to the axis seem to diverge from after passing through
the lens. The distance from the lens to this point is called the
focal length of the lens.

Because it seems rather odd to represent light as a dark line
on a white page, the diagram above has been inverted below to show
white light on a black background. The principle is the same.

Now the question is where one would find parallel light rays in
nature? How common or uncommon are parallel light rays if most of
the light we seen on a daily basis is diverging to one degree or
another?

If an object is very far away, the angle formed between
adjacent light rays is very small. Depending on the focal length
of the specific lens, this distance might be anywhere from a few
meters to a kilometer. If the object is very far, say 93,000,000
miles (1.5 x 1011 m) like the Sun, the distance is
sufficiently far that light rays are essentially parallel. So
sunlight is a convenient source of parallel light rays. Objects
that are a great distance away like hills or trees also furnish
rays that are almost parallel. Finally, lasers are a relatively
inexpensive source of parallel light due to their inherent nature.

NOTE: The light rays do not actually originate from the
focal point. Rather, their behavior on the other side of the lens
is such that they appear to be coming from there. Remember that
the original light rays were parallel to the axis!

NOTE 2: In the diagrams above, light rays are shown
bending at the center of the lens. This is a construction
technique and is used only for convenience. In fact the rays would
bend once upon entering the lens and a second time upon exiting.

BOTTOM LINE: If we see a light ray that's parallel to
the axis of a concave lens we know where it is going to go on the
other side -- it will diverge as if it had started at the focal
point.

DEFINITION 2:

Converging light rays striking a concave lens but headed
towards a point on the other side can be bent until they emerge
parallel to the axis. The point that causes this to happen is
called the focal point.

Or as before, white light on a black background:

NOTE: Because we have defined "focal point" so
precisely, we can understand that a light ray that is not parallel
to the axis will not diverge from the focal point on the other
side of the lens. Also we know that a light ray that does not head
towards the focal point will not emerge parallel to the axis.

BIG NOTE:

A concave lens has two focal points - one on each side. They
are equal distances from the lens. The lens does not have to have
the same curvature on both sides for this to be true, and it
doesn't depend on the direction the light takes entering the lens.
It is the combined curvature that determines the focal point.

BIGGER NOTE:

Because no light actually goes through the focal point of a
concave lens, it isn't "real" like the focal point of a convex
lens. Light is never focused there but only appears to come from
the focal point. The focal point of a concave lens is called
"virtual" which means that it only appears to have the effect of a
focal point.

When we purchase a concave lens, we specify the focal length
with a negative number such as f = -5 cm. When the mathematics of
image formation for concave lenses is worked out, it requires that
we use a negative number for the focal length to get a correct
answer.

HOW TO FIND THE FOCAL POINT:

If you wish to find the focal point of a concave lens, you
could take it outside on a clear day. Allow the sunlight to pass
through the lens and observe the pattern formed on a screen that
is parallel to the axis and located at the center of the lens.
IMPORTANT: Make sure the sunlight is aimed along the axis of the
lens.

The pattern of light on the opposite side of the lens will be
diverging. Trace at least two rays. You can place two pins in the
path of the light and trace their shadows. Project these two paths
back to where they intersect. This is the focal point. Remember,
originally parallel rays diverge as if coming from the focal
point.